Problem 27 - Calculus of a Single Variable

Question

Area The measurements of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectively. The possible error in each measurement is 0.25 centimeter. (a) Use differentials to approximate the possible propagated error in computing the area of the possible propagated (b) Approximate the percent error in computing the area of the triangle.

Step-by-Step Solution

Answer

The possible propagated error in computing the area of the triangle is 10.75 cm^2, and the approximate percent error in computing the area of the triangle is approximately 1.1944%.

Step 1: Derive Equation for Triangle Area

Start by setting up the function for the area of a triangle: \( A = 1/2 * base * altitude \), we know that base(B) = 36 cm and height(H) = 50 cm, thus \( A = 1/2 * B * H = 1/2 * 36 * 50 = 900 cm^2 \). Next, find the derivative (dA) of the area function A with respect to B and H. The Derivative of A w.r.t B, \(\frac{dA}{dB} = 1/2 * H = 1/2 * 50 = 25\) and the Derivative of A w.r.t H, \(\frac{dA}{dH} = 1/2 * B = 1/2 * 36 = 18\).

Step 2: Calculate Propagated Error Using Differentials

Now we have the differentials, to find the propagated error (∆A), sum the absolute values of the product of the differentials and the respective errors. Given maximum allowed error is \(∆B = ∆H = 0.25 cm\), we can sum up as follows: \(∆A = |dA/dB * ∆B| + |dA/dH * ∆H| = | 25 * 0.25| + | 18 * 0.25| = 6.25 + 4.5 = 10.75 cm^2\). Thus, the possible propagated error in computing the area of the triangle is 10.75 cm^2.

Step 3: Calculate Approximate Percent Error

Percent error can be calculated using the formula: Percent Error = ((|True Value - Experimental Value|) / True Value) * 100%. In this case it would be calculated as follows: Percent Error = (|∆A / A|) * 100% = ( |10.75 / 900 | ) * 100% = 1.1944%. Hence, the approximate percent error in computing the area of the triangle is approximately 1.1944%.

Key Concepts

Error Propagation

Error propagation is a useful concept that helps us understand how uncertainties in measurements affect our calculated results. When we know the possible errors in the measurements of a triangle's base and altitude, we can use differentials to estimate the propagated error in the area calculation.

To begin, we need the area formula for a triangle: \( A = \frac{1}{2} \, \text{base} \, \text{altitude} \). Here, the base is 36 cm and the altitude is 50 cm. The formula tells us that if there is a slight error in measuring the base or the altitude, it will directly impact the area.

Using calculus, specifically differentials, we derive the partial derivatives of the area with respect to the base (\(B\)) and the altitude (\(H\)). These derivatives tell us how much the area changes if the base or the altitude changes slightly.

The maximum variation in the area, known as error propagation, is then calculated by taking these derivatives and multiplying them by the possible change in base and altitude, each being 0.25 cm. This step-by-step process gives us a clear picture of how measurement imprecision "propagates" into our computed area.

Triangle Area

Calculating the area of a triangle is a foundational concept in geometry. It requires understanding the basic geometry formula \( A = \frac{1}{2} \, \text{base} \, \text{altitude} \). For this problem, we start with a base of 36 cm and an altitude of 50 cm.

To find the area, the formula takes half of the product of the base and altitude, yielding \( A = \frac{1}{2} \times 36 \times 50 = 900 \, \text{cm}^2 \). This formula tells us the space inside the triangle.

When the measurements are exact, the area is straightforward to calculate. However, due to potential measuring errors, the area might not be precisely 900 \(cm^2\). This is why understanding the role of potential measurement errors is crucial.

Triangles, as simple as they may seem, illustrate larger ideas about shapes and sizes, and they show how minor errors in measurements can affect a seemingly simple calculation like area.

Percent Error

Percent error is a critical concept that helps us measure the accuracy of our calculations compared to experimental results. It shows how much error exists relative to the size of the thing being measured, expressed as a percentage.

To calculate the percent error in our triangle's area, we find how the maximum possible error in area, \( \Delta A = 10.75 \, \text{cm}^2 \), compares to the calculated area of the triangle.

The formula for percent error is:

• Percent Error = \( \left( \frac{|\Delta A|}{A} \right) \times 100\% \)

Plugging in the numbers, we find Percent Error = \( \left( \frac{10.75}{900} \right) \times 100\% = 1.1944\% \).

This means the computed area of 900 \(cm^2\) could be off by about 1.1944\%. Understanding percent error not only helps us judge the reliability of this specific calculation but also builds foundational skills for assessing accuracy in all scientific and mathematical work.